Question: particular formula for sums of squares and square of sums

What is a general / particular formula (likely combinatorics or number theory) that describes a relationship between the sum of squares and square of sums? I remember once seeing a formula where “the ration between the sum of three numbers squared and the square of the sum of three numbers was ~1/3 or might have been ~2/3?

What is the name of this formula or a more general form of this?

$$1^2 + 2^2 + 3^2 +4^2 + ....+n^2 = \frac{n(n+1)(2n+1)}{6}$$
$$(1+2+3+4+...n)^2 = \left[\frac{n(n+1)}{2}\right]^2 = \frac{n^2(n+1)^2}{4}$$ So, $$\frac{1^2 + 2^2 + 3^2 +4^2 + ....+n^2}{(1+2+3+4+...n)^2} = \frac{n(n+1)(2n+1)}{6} \times \frac{4}{n^2(n+1)^2}$$ $$= \frac23 \left(\frac{2n+1}{n(n+1)}\right)$$